DISTORTION analysis has gained renewed interest because

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH Disttion in Elementary Transist Circuits Willy Sansen, Fellow, IEEE Abstract In this paper the disttion components are defined f elementary transist stages such as a single-transist amplifier a differential pair using bipolar transists MOST s. Meover, the influence of feedback is examined. Numerical examples are given f sake of illustration. Index Terms Amplifiers, disttion, feedback, intercept point. I. INTRODUCTION DISTORTION analysis has gained renewed interest because it is responsible f the generation of spurious frequency bs in telecommunication circuits. Therefe, it is reviewed starting with the most elementary circuit blocks [2], [4] [6]. Disttion actually refers to the disttion of a voltage current wavefm as it is displayed versus time, i.e., as seen on a oscilloscope. Any difference between the shape of the output wavefm versus time the input wavefm, except f a scaling fact, is called disttion. F example, the flattening of a sinusoidal wavefm is disttion. The injection of a spike on a sinusoidal wavefm is called disttion as well. Several kinds of disttion occur. They are defined first. Fig. 1. Application of a high-pass filter causes linear disttion because of the reduction of the low frequencies. Fig. 2. Application of a low-pass filter causes linear disttion because of the reduction of the high frequencies. A. Linear Nonlinear Disttion Linear disttion is caused by the application of a linear circuit, with a nonconstant amplitude phase characteristic. As an example, the application of a high-pass filter (of first der) to a square wavefm causes disttion, as shown in Fig. 1. In a similar way, the application of a low-pass filter reduces the high-frequency content in the output wavefm, as shown in Fig. 2. Nonlinear disttion is caused by a nonlinear transfer characteristic. F example, the application of a sinusoidal wavefm to the exponential characteristic of a bipolar transist causes a sharpening of one top flattening of the other one (see Fig. 3). This cresponds to the generation of a number of harmonic frequencies of the input sinusoidal wavefm. These are the nonlinear disttion components. B. Weak Hard Disttion When the nonlinear transfer characteristic has a gradual change of slope (as shown in Fig. 3), then the quasi-sinusoidal wavefm at the output is still continuous. This is not the case when the transfer characteristic has a sharp edge, as shown in Fig. 4 f a class B amplifier. Part of the sinusoidal wavefm Manuscript received July 31, 1997; revised June 15, This paper was recommended by Guest Edit A. Rodriguez-Vazquez. The auth is with ESAT-MICAS, K.U. Leuven, Leuven, Belgium. Publisher Item Identifier S (99) Fig. 3. Generation of nonlinear disttion caused by the nonlinear i C v BE characteristic. is then simply cut off, leaving two sharp cners. These cners generate a large number of high-frequency harmonics. They are sources of hard disttion. In the case of weak disttion, the harmonics gradually disappear when the signal amplitude becomes smaller. They are never zero, however. They can easily be calculated from a Tayl series expansion around the quiescent operating /99$ IEEE

2 316 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1999 Fig. 5. Disttion components versus nmalized input voltage. Fig. 4. Generation of hard nonlinear disttion. point, as will be carried out in next paragraph. Hard disttion harmonics, on the other h, suddenly disappear when the amplitude of the sinusoidal wavefm falls below the threshold, i.e., the edge of the transfer characteristic. Also they are much me difficult to calculate. Since they can be avoided altogether by limiting the output signal amplitudes to sufficiently low levels, they will not be discussed any further. In this paper, the nonlinear disttion will be calculated f elementary bipolar MOST amplifier buffer stages. Also the influence of negative feedback is derived. First, however, the several definitions have to be reviewed to describe the weak nonlinear disttion components. II. WEAK-DISTORTION COMPONENTS Let us consider an amplifier with a weak nonlinearity as in Fig. 3. Both input output signals vary with time. They are denoted by, in shth,. At low frequencies, the output of this amplifier can be expressed in terms of its input by a power series Coefficient represents the dc component of output signal. Coefficient represents the linear gain of the amplifier, whereas coefficients, represent its disttion. Coefficients, can be obtained from the analytic expression of the function as given by Application of a cosine wavefm of frequency amplitude at the input of that amplifier yields output components at all multiples of. It is obtained by trigonometric manipulation. Under low-disttion conditions, only second- third-der disttion components are considered. By use of the expressions (1) (2) (3) (1) thus becomes Odd-der disttion, especially, thus modifies the signal component at the fundamental frequency. Term can be neglected, however, with respect to, provided the signal amplitude is sufficiently small. Harmonic disttion is then defined as follows. The th harmonic disttion (HD ) is defined as the ratio of the component of frequency to the one at the fundamental. Application to (4) yields HD HD (4) (5a) (5b) It is imptant to note that HD is proptional to HD to. Increasing the input signal level by 1 db thus increases the HD by 1 db the HD by 2 db. These relationships hold true f all values of, which are not too large. This is the region where the so-called low-disttion conditions are valid. F even larger values of, the values of HD HD flatten off with increasing as shown in Fig. 5. In this paper, the analyzes are limited to the region of low disttion, i.e., where is sufficiently small, i.e., where HD is proptional to HD to. Also the total harmonic disttion THD is given by THD HD HD (5c) It is not very useful as it does not give a clear dependence on the input signal level.

3 SANSEN: DISTORTION IN ELEMENTARY TRANSISTOR CIRCUITS 317 Application of the sum of two cosine wavefms of frequencies both of amplitude at the input gives rise to output signal components at all combinations of, their multiples. Under low-disttion conditions, the number of terms can be reduced to the ones caused by coefficients only. They are mapped versus frequency in Fig. 6(a) f (10 MHz) (11 MHz). A real frequency spectrum f frequencies MHz is shown in Fig. 6(b). Second-der intermodulation disttion ( ) is then defined by the ratio of the component at frequency to the one at. Under low-disttion conditions (6a) Third-der intermodulation disttion ( ) can be detected at the frequencies [see Fig. 6(a)]. It is given by the ratio of the component at frequency ( one of the other three frequencies), which is, to the fundamental, which is, as given by (6b) Comparison of the four equations above shows that HD (7a) HD (7b) Under low-disttion conditions, there is thus a one-toone crespondence between harmonic intermodulation disttion. It is thus sufficient to specify only one of them. Note that two of the four equal components, i.e., the ones at the frequencies, occur closely to the two fundamentals. This is one reason why they are me imptant than the HD components. In music signals f instance, it is quite conceivable that two peaks which are close together in frequency, generate intermodulation products in the same frequency range. At high frequencies, these products may already be reduced by the amplitude-frequency characteristic. In Fig. 6(b), the peaks are clearly visible at frequencies MHz. The is thus about 40 db. The other two components around 30 MHz are already heavily attenuated (not in the picture). A second reason why the measurement of is preferred above the one of HD is that the value of is three times larger than the one of HD hence easier to measure. F these reasons, the value of is always preferred. Another imptant characteristic often used point is the intercept,. It is the value of the input signal where the extrapolated curves of the components of the fundamental coincide. This is shown in Fig. 7. The output components at the fundamental frequencies at the frequencies are plotted versus the input voltage. They are given by, respectively, (note that is again dimensionless but that has as dimension). is the ratio of both components. The point where both components coincide is. It is thus also the point where (a) (b) Fig. 6. (a) Second- third-der harmonic intermodulation components. (b) Intermodulation disttion of a 10.7 MHz filter [3].. This point is easy to calculate from (6b) is given by (8) Obviously, the smaller, the larger the value of. Another related measure f the disttion is the Intermodulation free dynamic range (FDR ). The dynamic range is

4 318 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1999 db Fig. 7. Fundamental 3 components versus input voltage. the ratio of the maximum output signal to the output noise, as shown in Fig. 7. It is thus given by DR DR (9) in which is the input noise (in ). The FDR is the largest possible DR without disttion. It is thus obtained at the input voltage where the output noise equals the component (see Fig. 7), where The difference between both is thus almost 10 db. The measurement of the 1 db compression point is thus an easy way to obtain the value of. There are several other ways to describe the disttion caused by coefficients such as cross-modulation disttion, triple beat, etc. There is nevertheless always a constant relationship of the type (7) between them. Therefe, only one me disttion is shtly discussed. It is the cross-modulation disttion. F the determination of cross-modulation disttion, again, two carrier frequencies are required. The first one, however, is modulated by a modulating signal at low frequency. The modulation index, which is ratio of the amplitude of the modulating signal to the one of the carrier, is denoted by. A nonlinear transfer characteristic causes the modulation to be transferred from the first carrier to the other one. As a result, the second carrier is modulated as well. This causes mixing of the channels in cable TV, etc., is thus to be avoided. The modulation index of the other channel is a measure of the disttion, is called the cross-modulation disttion. It is given by which yields Substitution of this value in (9) finally gives FDR (10a) CM (12a) CM (12b) Note that CM is only generated by the third-der terms of the power series, which describe the nonlinearity. Since it is closely related to, it will not be discussed any further. FDR (10b) III. DISTORTION IN A BOLAR TRANSISTOR AMPLIFIER In a bipolar transist, the collect current is controlled by the base emitter voltage as given by FDR An alternative, albeit less accurate, way to characterize disttion is the 1 db compression point (see Fig. 7). It is the value of where the fundamental component is compressed by 1 db, is denoted by. This value can be approximately calculated from (4). Indeed, the compression is caused by the second term (in ) of the coefficient of. A reduction of 1 db is a reduction to The resultant value of is thus about given by (11) (13) in which is the collect saturation current (see [1, Ch. 1]) mv at 29 C ( 302 K). The transist is biased at a specific dc value of, i.e., in quiescent point of the characteristic (see Fig. 3). A small variation of this voltage causes a variation in collect current. These variations ac components of the collect current the base emitter voltage can be expressed as given by (14)

5 SANSEN: DISTORTION IN ELEMENTARY TRANSISTOR CIRCUITS 319 Expression (13) thus results in (15) After division of both terms by the value of the quiescent current, we obtain (16) with, which is called the relative current swing. It is the current variation in the transist, nmalized to the quiescent dc current. It is a measure of the fraction of the dc current in the transist, which is used to generate ac output signal. It will be used throughout this section to compare disttion perfmance. F small peak base emitter voltages, the exponential of (16) can be exped in a Tayl series. Indeed, f, we know that application of this expansion to (16) yields (17) (18) in which is the peak value of the relative current swing, is the peak value of the ac base emitter input signal. F small input signals, only the first term in (18) has to be retained, which leads to (19) which is well expected. Meover, in first der, the peak value of the relative current swing is derived from the peak input voltage as given by (20) Finally, identification of (18) with (1) shows that f, the coefficients are,,,. Use of the (5) (10) substitution of by as given by (20) yields HD (21a) HD (21b) F example, a peak ac current of 100 A in a bipolar transist, carrying 1 ma, causes a peak relative current swing of, % ( HD %), also % ( HD %). F this ac current, a peak input voltage is required of only 2.6 mv to 1.84 mv. A larger peak current swing of 0.5 leads to %, f which an input signal amplitude of 9.2 mv is sufficient. F k, the voltage gain then equals 200. Finally, the value of the on the scale of the current swing (22) on the input voltage scale. This cresponds with an input voltage of 73 mv. This is quite small. A bipolar transist with 1 ma has a ms. With a base resist of, its equivalent input noise is the noise of (see [1]). This cresponds with 1.56 nv Hz. F a bwidth of 200 khz, the noise level V. As a result FDR 67 db. It is imptant to note that disttion components can always be described by means of the input voltage drive by the current swing. The latter way has a number of advantages. The current swing already includes the effects of the transconductance of the feedback such that the expressions become simpler very much comparable. They will be used throughout this paper. From these numbers, it is clear that only small input signal amplitudes can be applied to a bipolar transist. Also, a current swing of 0.5 already cresponds with a high disttion region, as shown in Fig. 5. F small values of, relations (21) (22) hold. On a double logarithmic scale, straight lines result with slopes of 1 2, respectively. Doubling thus quadruples the thirdder disttion. At higher values of, however, the values of the disttion are quite high but do not increase (see Fig. 5) any further. These values have been calculated by means of transient analyzes in SPICE, followed by Fourier analyses. IV. DISTORTION IN A MOSFET AMPLIFIER F a MOST, the analysis is very similar as f a bipolar transist. Only the transfer characteristic is quadratic not exponential. Less disttion is thus expected. The drain current gate source voltage of a MOST are in first-der related by (23) in which is the transconductance fact, which includes the size, is the threshold voltage. The transist is biased at a specific dc value of, i.e., in a quiescent operating point. A small variation of this voltage causes a small variation in drain current. They are related by Expression (23) thus becomes (24) (25)

6 320 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1999 Subtraction of from both sides, division by the value of the quiescent current, yields (26) (27) in which are the peak values of the relative current swing the gate source input voltage, respectively. F small signals, only the first term in (27) has to be retained, which yields (28) as expected. Also, the peak relative current swing is related to the input drive by (29) Finally, identification of (27) with (1) shows that f, the coefficients are,,,. Use of definitions (5) (7) thus yields HD (30) (31) Note that no third-der disttion occurs. Indeed, the transfer characteristic [expression (23)] is only quadratic, hence no third-der terms can be generated. Hence, is zero infinite. Comparison of (30) with (21) shows that a MOST only generates half as much (second-der) disttion as a bipolar transist. The main advantage of a MOST, however, is that the input voltage is scaled to ( ), which can be made quite large, whereas f a bipolar transist, the input voltage is fixed scaled to mv. F example, if again a peak relative current swing is taken of 0.1 (f ma 100 A peak ac current), then %. Even me imptant, however, is that a peak input voltage is allowed of mv (35 mv ) f V, of 10 mv (7 mv ) only, f V. The smaller the aspect ratio is made, the larger the value the larger the peak input voltage can be allowed f the same disttion. The input voltage is indeed related to the disttion ( the relative current swing )as given by (32) Fig. 8. Generation of nonlinear disttion (compression), caused by a symmetrical differential stage. F a given amount of disttion ( ) dc current ( ), the maximum value of is inversely proptional to the square root of hence,. Finally note that no third-der disttion is generated as long as the first-der model of a MOST is guaranteed. As soon as the complete expression is taken of MOST, including the terms with 3/2 exponents, then third-der disttion does occur, but nevertheless in very limited amounts. V. DISTORTION IN A BOLAR TRANSISTOR DIFFERENTIAL AMPLIFIER Phase inversion of the input signal changes the sign of the fundamental third-der components but not of the secondder component. This is exploited in a balanced differential circuit, to which two input signals of equal amplitude but opposite phase are applied. The difference of the output signals does not contain even-der disttion at least if no unbalance is caused by mismatch. This is the case f a differential amplifier as discussed next. As shown by the transfer characteristic (see Fig. 8), the operating point occurs now at zero output input voltage. The transfer characteristic is indeed perfectly symmetrical with respect to the crosspoint of the axis. Application of a sinusoidal wavefm in causes a flattening of both tops of the quasi sinusoidal wavefm in. Compression thus occurs. The transfer characteristic has been derived in [1, Ch. 4]. The differential output current is twice the ac current in each transist. The relative current swing is thus given by (33) in which is the ac current circulating through both transists is the differential input voltage. If two load resists were added, then the output voltage would be. F small input voltages ( ), the tanh function can be exped in a power series. Indeed, f, we know that (34)

7 SANSEN: DISTORTION IN ELEMENTARY TRANSISTOR CIRCUITS 321 Application to (33) yields (35) in which represent peak values of the relative current swing the input voltage, respectively. Truncation of this power series after its first term is sufficient an approximation f small signals. It leads to the well-known result that (36) in which is the transconductance of both transists, both carrying current. In a first-der approximation, a simple relation is also obtained between the input voltage the relative current swing, as given by (37) Finally, identification of (35) with (1) shows that f, the coefficients are,,,. Use of (5) (10) of relation (37) yields as expected Also, HD (38a) HD (38b) (39) Coefficient is negative, hence, the disttion causes compression of the wavefm. F example, a total dc current ma is used again. Now, however, each bipolar transist only carries a dc current of 0.5 ma. The peak ac current in each transist is also reduced to 50 A. F k, the voltage gain also equals 200. The peak relative current swing is again. As a result, %. F this, a peak input voltage is obtained of 5.2 mv 3.7 mv. The disttion is thus 2 times lower than in the case of a single transist carrying a dc current providing the same gain. This fact of 2 is also found by comparison of (38) with (21). This conclusion is especially true because no second-der disttion is present. In practice, mismatch will generate some second-der disttion as well. It is usually much smaller than the third-der disttion. F a peak relative current swing of 0.5, % f which a signal amplitude of 8.4 mv is required. Again, a fact of 2 difference is found. It can be concluded that a differential stage can take 1.4 times me input voltage to generate the same third-der disttion as a single transist amplifier with the same total dc current. VI. DISTORTION IN A MOST DIFFERENTIAL AMPLIFIER The transfer characteristic of a differential pair with MOST is very similar to the one with bipolar transists; it is symmetrical around the igin. No second-der disttion can thus occur. Since a single MOST amplifier does not generate third-der disttion, it will be interesting to examine what disttion perfmance can be obtained with a MOST differential amplifier. The transfer characteristic has been derived in [1, Ch. 4]. The differential output current is again twice the ac current in each transist. The relative current swing is thus given by (40) in which is the differential input voltage. Note that can always be substituted by. F small values of, the square root can be exped as a power series. Indeed, f, we know that which allows us to wk out (40) into (41) (42) Again,,, all represent peak values. F a pure small signal analysis, the power series has to be limited to the first term only, which leads to (43) (44) in which is the gate source voltage, is the transconductance of either T1 T2, which both carry currents of. Expression (42) also provides a first-der relation between the input voltage the relative current swing (45) In der to obtain the disttion components, (42) has to be identified with (1). It shows that f, the coefficients are,,,. No second-der disttion thus occurs, as expected indeed, since no quadratic component occurs in (42). Also, coefficient is negative, which shows that compression disttion occurs, as expected as well, from a differential stage. Use of the definitions (5) (10) of relation (45) yields zero f HD (46a)

8 322 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1999 in which represents the transfer function of the unilateral feedback netwk. The coefficients of the new power series can be found by application of (2) on (1) use of (48), which yields the following relations (49) (50) (51) Fig. 9. Application of negative feedback f converts power series coefficients a i into b i. (46b) F example, both transists carry a dc current of 0.5 ma a peak ac component of 50 A A, which yields. As a result %. This value is 1.6 times larger than the one f a bipolar differential stage with the same current swing. However, the input voltage allowed, again depends on the value of as given by (45), which can be rewritten as (47) in which applies to either transist T1 T2. The smaller ( ), the larger the input voltage allowed. F example, if V, then mv 14 mv. Finally, V. It can thus be concluded that a MOST differential stage does generate third-der disttion, because of the limiting action of its transfer characteristic. It even generates a somewhat me third-der disttion than a bipolar differential stage. The input voltage allowed is, however, much larger can be designed to, in principle, any value, depending on the value of. VII. THE EFFECT OF FEEDBACK ON DISTORTION Series base emitter resistances in the bipolar transist linearize the exponential relationship thus reduce the disttion. This cresponds, however, with a reduction in gain. Also, series source resistance in the MOST reduces the disttion the gain as well. In this section, it is examined how the application of negative feedback reduces the disttion. A. They The application of negative feedback around the nonlinear amplifier, which is characterized by coefficients (see Fig. 9) gives rise to a new power series of the same fm, but with coefficients. The feedback action is described by (48) in which the loop gain is given by (52) All expressions (5) (10) are used to obtain the disttion components are still valid, provided the coefficients are replaced by. The amplitude of the output signal itself is given by (49). It is reduced by a fact of as expected. F this reason, the input voltage (see Fig. 9) is reduced by as well. The second-der disttion is given by (53) Also, after replacement of by (54) The first term represents third-der disttion related to, which is present as well without feedback. It is positive thus represents expansion disttion. A sinusoidal wavefm becomes me triangular. The second term represents second-der interaction around the feedback loop, generating third-der disttion. It is negative thus cresponds with compression. The third-der disttion can cancel completely f specific values of. This causes a null in the characteristic, which is quite sharp difficult to maintain over a wide range of transist variables. Therefe, it is never a parameter to design f. Meover, it occurs at very small values of loop gain. F high values of, the second term usually dominates compression disttion results. F small values of of, expansion disttion is dominant. These effects are now illustrated with several examples. B. Emitter Resistance in Single Bipolar Transist Amplifier Insertion of an emitter resistance in a single transist amplifier provides local feedback. The loop gain is given by (55) The second-der disttion component is then obtained from (53) given by ( f a bipolar transist) (56)

9 SANSEN: DISTORTION IN ELEMENTARY TRANSISTOR CIRCUITS 323 which shows that the input voltage is to be compared with the voltage drop across the feedback resistance, in der to obtain. F instance, f a voltage drop across of 1 V ( k with ma), then V 0.07 V gives. F such high values of feedback ( ), the disttion components can be simplified to (61) (62) Fig. 10. Disttion components with feedback in a bipolar transist with 1 ma collect current. in which is the peak input voltage with respect to ground. Also since the result is (57) (58) The third-der disttion is derived from (54) given by ( ; ) (59) F example, a bipolar transist carries a dc current of 1 ma an ac peak current of 100 A. The peak relative current swing is thus 0.1. Without feedback %, %, mv. Addition of a resistance of 260 causes a dc voltage across of mv which results in. Note that the value of is easily found by taking the dc voltage across, divided by. The value of %. Meover, the input voltage allowed increases to 20.2 mv. The value of is then, 0.02%. In der to increase to the same value as without feedback, the value of has to be increased by 2.5, yielding mv. The disttion components with feedback are plotted versus ( ) in Fig. 10 f constant values of the collect current ( ma) relative current swing. F low feedback ( ), the values are the same as on Fig. 5 f. F large feedback ( ), the values decrease with a slope of unity. Note that the null in indeed occurs at, which cresponds with a very small amount of feedback indeed. F high values of feedback ( ), expression (57) of the relative current swing can be modified into (60) Comparison with (21) shows that f it is sufficient to divide by, whereas has to be divided by. It can thus be concluded that feedback reduces disttion components indeed. All of them are reduced, however, by about the same amount. Finally, note that emitter resistances can never fully be excluded in a bipolar transist since the base resistance linearizes the exponential as well. The equivalent emitter resistance is then, which is usually of the der of a few ohms. C. Source Resistance in Single MOST Amplifier The insertion of a source resist provides local feedback. The value of the loop gain is again given by (55) with an emitter resist instead of a source resist. From (27), we find. As a result, (53) (54) become (63) since now (64) (65) F the same current swing, the second-der disttion is reduced by. Now, however, third-der disttion emerges as well. It is caused by the presence of in (54), which represents the increase in der of the second-der disttion component which is fed back to the input. It is still smaller than f a bipolar transist. F large feedback, the current swing becomes (66) (67) which leads to the same conclusion as f a bipolar transist.

10 324 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1999 D. Emitter Resistances in a Bipolar Differential Stage In a differential pair, second-der disttion is absent ( ). Addition of equal emitter resistances in both transists does not degrade this symmetry. The third-der disttion is derived from (54) is given by (f ) (68) The same conclusions can thus be drawn. is negative, which cresponds with compression disttion, as befe. F large feedback ( ), the value of decreases linearly with is then given by the disttion without feedback, divided by ( ). E. Source Resistances in a MOST Differential Stage Again, symmetry is maintained, hence no second-der disttion occurs. From (42) we find that. The third-der disttion is again derived from (54) is given by (69) The same conclusion can be drawn as f a differential stage with bipolar transists. F. Emitter Follower F disttion analysis, the emitter follower can be regarded as a single transist amplifier with large feedback ( ). The output is taken at the emitter instead of at the collect; but since the relative current swing is taken as a fundamental parameter, the analysis is the same. F an emitter follower with an emitter resistance, the disttion components are thus already given by (61) (62). However, if a transist is used instead of a resistance, then its output resistance has to be used in the expression instead of. Since, in which is the early voltage, the relative current swing can be derived from (60) is given by (70) In der to obtain, the input voltage thus simply has to be compared with the early voltage. F instance, f V( ma), an input voltage of V ( 0.07 V ) only provides. The disttion components are then given by (61) (62) which give ( ) % %. They are thus negligible, thanks to both the low values of the high value of. F an ideal follower, the current source is ideal, its current is not modified by application of an input signal. Hence, the current swing is zero so is the disttion (see Fig. 11). The disttion of a source follower can also be calculated directly as a solution of a nonlinear equation. Fig. 11. The current swing in an ideal source follower is zero, so is the disttion. G. Source Follower Very much the same conclusions apply to the source follower as to the emitter follower. The relative current swing is again given by (71) has to be used in (63) (67). As an example, a source follower is taken at ma with a current source with output resistance 16 k ( V). An input voltage of 4 V ( 2.8 V ) now gives. Now the aspect ratio is such that V. V. Thus, % %. Obviously f an ideal current source, the relative current swing is zero so is the disttion (see Fig. 11). In this consideration, the bulk is assumed to be connected to the source. If this is not the case, the parasitic JFET the body effect has to be considered as well. In this case, the disttion is mainly caused by this effect. To find the sources of disttion in any arbitrary circuit, the values of the relative current swing have to be found together with the feedback fact. All disttion components are readily calculated. In addition, the amplitude of the transfer characteristic versus frequency has to be calculated of each transist output to the output of the total circuit. Higher harmonics are usually attenuated by the low-pass filter action of the capacitances present. REFERENCES [1] K. Laker W. Sansen, Design of Analog Integrated Circuits Systems. New Yk: McGraw-Hill, [2] W. Sansen R. Meyer, Disttion in bipolar transist variable-gain amplifiers, IEEE J. Solid-State Circuits, vol. SC-8, pp , Aug [3] J. Silva-Martinez, M. Steyaert, W. Sansen, High-Perfmance CMOS Continuous-Time Filters. Nwell, MA: Kluwer Academic, [4] S. Willingham K. Martin, Integrated Video-Frequency Continuous- Time Filters. Nwell, MA: Kluwer, [5] D. Pederson K. Mayaram, Analog Integrated Circuits f Communications. Nwell, MA: Kluwer, [6] P. Wambacq W. Sansen, Disttion Analysis of Analog Integrated Circuits. Nwell, MA: Kluwer, 1998.

11 SANSEN: DISTORTION IN ELEMENTARY TRANSISTOR CIRCUITS 325 Willy Sansen (S 66 M 72 SM 86 F 95) received the M.Sc. degree in electrical engineering from the Katholieke Universiteit Leuven in 1967 the Ph.D. degree in electronics from the University of Califnia at Berkeley in Since 1981, he has been a Full Profess at the ESAT Labaty of the Katholieke Universiteit Leuven. He was a Visiting Profess at the Universities of Stanfd (1977), Lausanne (1981), Philadelphia (1985), Ulm (1994). He has been involved in design automation in numerous analogue integrated circuit designs f telecom, consumer, biomedical applications senss. He has been supervis of 340 papers in international journals conference proceedings six books, among which the textbook with K. Laker, Design of Analog Integrated Circuits Systems (McGraw- Hill, 1994).